I recently came across Sukkah 55a, which lists the Shirei Shel Yom for Chol HaMoed Sukkos. Are there any other Gemaras which discuss the Shirei Shel Yom of other Yamim Tovim? What are they? Are there any that we have a tradition for which are not in the Gemara?

Traditionally in standard Ashkenazi Shuls (excluding those that follow the Gra there is a link to his list in Joelks comment) we don't say any Mizmor other than the Daily ones, except on Rosh Chodesh where we add Tehillim 104, Chanuka Tehillim 30 (mentioned above from source).
It happens to be that on Rosh Hashana we do say Tehillim 47 mentioned above before blowing the Shofar, and on Yom Kippur we say Tehillim 130 before Bircos kerias Shema mentioned above,Tehillim 135 mentioned above on Rishon Shel Pesach during Pesukei Dezimra, and Tehillim 136 mentioned above on Acharon Shel Pesach during Pesukei Dezimra.